\section{1.17} 
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1.17. Support of a $\mathcal{D}$-module. 

The support $\mathrm{Supp}(F)$ of $F \in \mu(\mathcal{D}_X)$ is, as usual, the set of $x \in X$ for which $F_x \ne 0$. 

If $F \in \mathrm{coh}(\mathcal{D}_X)$, it is closed, since the support of a section of a sheaf of abelian groups is always closed. 

Let $\sigma: T^*(X) \to X$ be the canonical projection. 

Then $\mathrm{Supp}\,F = \sigma(\mathrm{SS}(F))$. 

In particular $\sigma(\mathrm{SS}(F))$ is closed. 

This can also be seen directly: Let $Z$ be an irreducible component of $\mathrm{SS}(F)$. 

If $Z \subset X$, then its image is clearly closed. 

If not, $Z$ is conical (1.9(6)), hence the restriction of $\sigma$ to $Z$ factors through the projection onto $X$ of the projectivised cotangent bundle, which is a proper map, and therefore sends $Z$ onto a closed set.

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